Supporting functional

In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

Let X be a locally convex topological space, and

be a convex set, then the continuous linear functional

ϕ :

is a supporting functional of C at the point

ϕ ≠ 0

ϕ ( x ) ≤ ϕ (

is the dual space of

) is a support function of the set C, then if

defines a supporting functional

ϕ :

of C at the point

ϕ ( x ) =

ϕ

is a supporting functional of the convex set C at the point

ϕ

( σ )

defines a supporting hyperplane to C at